Problem 27 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{8}$$

Step-by-Step Solution

Answer

-3

Step 1: Identify Base and Number

Here, the base is 2 and the number is \( \frac{1}{8} \). In other words, we need to find the power to which 2 should be raised to produce \( \frac{1}{8} \).

Step 2: Express Number in terms of Base

The number \( \frac{1}{8} \) can be written as \( 2^{-3} \), since \( 2^{-3} = \frac{1}{8} \).

Step 3: Apply Definition of Logarithm

The definition of a logarithm states that if \( a^b = c \), then \( \log _{a} c = b \). So, applying this definition here, we get \( \log _{2} 2^{-3} = -3 \).